kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2026-23-1-265-280kaz29-2520Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESВАРИАЦИОННЫЙ ПОДХОД К ИДЕНТИФИКАЦИИ ПАРАМЕТРОВ СРЕДЫ ПОСРЕДСТВОМ ОБРАТНОГО АНАЛИЗА РАСПРОСТРАНЕНИЯ АКУСТИЧЕСКИХ ВОЛНVARIATIONAL APPROACH FOR IDENTIFYING ENVIRONMENTAL PARAMETERS VIA INVERSE ANALYSIS OF ACOUSTIC WAVE PROPAGATIONhttps://orcid.org/0000-0002-5828-7820СиницаА. В.SinitsaA. V.,

PhD, ассистент-профессор

г. Алматы

PhD, Assistant Professor

Almaty

a.sinitsa@kbtu.kzhttps://orcid.org/0009-0003-0402-4720ШкоркоA. K.ShkorkoA. K.

магистр, лектор

г. Алматы

MSc, Lecturer

Almaty

a.ukassova@kbtu.kzhttps://orcid.org/0009-0003-0772-4319ЦхайЮ. A.TskhayY. A.,

магистр, лектор

г. Алматы

MSc, Lecturer

Almaty

y.tskhay@kbtu.kzhttps://orcid.org/0000-0002-8142-6710КардукА. Р.KarduckA. P.

PhD, профессор

г. Фуртванген

PhD, Professor

Furtwangen

achim.karduck@hs-furtwangen.deКазахстанско-Британский технический университетКазахстанKazakh-British Technical UniversityKazakhstanУниверситет прикладных наук ФуртвангенаГерманияHochschule Furtwangen UniversityGermany202629032026231265280Copyright © Синица А.В., Шкорко A.K., Цхай Ю.A., Кардук А.Р., 20262026Синица А.В., Шкорко A.K., Цхай Ю.A., Кардук А.Р.Sinitsa A.V., Shkorko A.K., Tskhay Y.A., Karduck A.P.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/2520

В статье представлен численный метод восстановления пространственного распределения скорости звука в неоднородных средах на основе обратного анализа распространения акустических волн. Математическая модель базируется на волновом уравнении второго порядка с переменными коэффициентами. Обратная задача сформулирована как задача оптимизации по минимизации функционала невязки между смоделированными и наблюдаемыми данными давления на границах области. Для эффективного вычисления градиента функционала применяется метод сопряженной (вспомогательной) задачи, выведенный с помощью вариационного исчисления. Численная реализация выполнена с использованием явной конечно-разностной схемы. Вычислительные эксперименты на одномерной модели гетерогенной среды (грунт – металл – грунт) показывают, что предложенный алгоритм позволяет достоверно восстанавливать профиль скорости, особенно в зонах резкого контраста. В работе проведен анализ чувствительности решения и скорости сходимости, показавший, что 500 итераций обеспечивают оптимальный баланс между точностью и вычислительными затратами.

This paper presents a numerical method for reconstructing the spatial distribution of sound speed in inhomogeneous media based on the inverse analysis of acoustic wave propagation. The mathematical model relies on the second-order wave equation with variable coefficients. The inverse problem is formulated as an optimization task to minimize the residual functional between simulated and observed pressure data at the domain boundaries. To efficiently calculate the gradient of the functional, an adjoint (auxiliary) problem method is employed, derived via variational calculus. The numerical implementation is performed using an explicit finite-difference scheme. Computational experiments on a one-dimensional model of a heterogeneous medium (soil-metal-soil) demonstrate that the proposed algorithm allows for reliable reconstruction of the velocity profile, particularly in zones of sharp contrast. The study analyzes the sensitivity of the solution and the convergence rate, showing that 500 iterations provide an optimal balance between accuracy and computational cost.

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The authors declare that there are no conflicts of interest present.